16.3: Odd Clock

Consider the clock depicted in the following image.

Image from: Hackaday.

Instead of a standard clock–which has independent hour and minute hands–this clock connects the minute hand at the end of the hour hand. Here is a video showing the sped-up clock motion:

from IPython.display import YouTubeVideo
YouTubeVideo("bowLiSlm_gA",width=640,height=360, mute=1)

The following code is an animated traditional clock which uses the function as a trick to animate things in jupyter:

%matplotlib inline
import matplotlib.pylab as plt
from IPython.display import display, clear_output
import time
def show_animation(delay=0.01):
    fig = plt.gcf();
    time.sleep(delay)       # Sleep for half a second to slow down the animation
    clear_output(wait=True) # Clear output for dynamic display
    display(fig)            # Reset display
    fig.clear()             # Prevent overlapping and layered plots

Lets see a standard analog clock run at high speed

import numpy as np
'''
Analog clock plotter with time input as seconds
'''
def analog_clock(tm=0):

    #Convert from time to radians
    a_minutes = -tm/(60*60) * np.pi * 2
    a_hours = -tm/(60*60*12) * np.pi * 2

    #Define clock hand sizees
    d_minutes = 4
    d_hours = 3 
    arrow_width=0.5
    arrow_length=1

    # Set up figure
    fig = plt.gcf()
    ax = fig.gca();
    ax.set_xlim([-15,15]);
    ax.set_ylim([-10,10]);
    ax.scatter(0,0, s=15000, color="navy"); #Background Circle
    plt.axis('off');
        
    # Calculation Minute hand transformation matrix
    J2 = np.matrix([[np.cos(a_minutes), -np.sin(a_minutes)], 
                    [np.sin(a_minutes), np.cos(a_minutes)]] )
    pm = np.matrix([[0,d_minutes], [-arrow_width,d_minutes], [0,arrow_length+d_minutes], [arrow_width,d_minutes], [0,d_minutes]] ).T;
    pm = np.concatenate((J2*pm, np.matrix([0,0]).T), axis=1 );
    ax.plot(pm[0,:].tolist()[0],(pm[1,:]).tolist()[0], color='cyan', linewidth=2);

    # Calculation Hour hand transformation matrix    
    J1 = np.matrix([[np.cos(a_hours), -np.sin(a_hours)], 
                    [np.sin(a_hours), np.cos(a_hours)]] )
    ph = np.matrix([[0,d_hours], [0,d_hours], [-arrow_width,d_hours], [0,arrow_length+d_hours], [arrow_width,d_hours], [0,d_hours]]).T;
    ph = np.concatenate((J1*ph, np.matrix([0,0]).T), axis=1 );
    ax.plot(ph[0,:].tolist()[0],(ph[1,:]).tolist()[0], color='yellow', linewidth=2);

#Run the clock for about 5 hours at 100 times speed so we can see the hands move
for tm in range(0,60*60*5, 100):
    analog_clock(tm);
    show_animation();
# 'Run' this cell to see the animation

For the following few questions, consider the transformation matrix \(J_1\) redefined below with an angle of 5 hours out of 12.

import sympy as sym
import numpy as np
sym.init_printing(use_unicode=True)

a_hours = 5/12 * 2 * np.pi
J1 = np.matrix([[np.cos(a_hours), -np.sin(a_hours)], 
                [np.sin(a_hours), np.cos(a_hours)]] )

sym.Matrix(J1)

Question

Using code, show that the transpose of \(J_1\) is also the inverse of \(J_1\), then explain how the code demonstrates the answer.

#Put your answer here

Question

Given the trigonometric identity \(cos^2(\theta) + sin^2(\theta) = 1\), prove by construction–using either Python or LaTeX/Markdown or sympy (if you are feeling adventurous)–that the transpose of the \(J_1\) matrix is also the inverse for ANY angle a_hours \(\in[0,2\pi]\).

Now consider the following code which attempts to connect the hands on the clock together to make the Odd Clock shown in the video above.

import numpy as np

def odd_clock(tm=0):

    #Convert from time to radians
    #a_seconds = -tm/60 * np.pi * 2
    a_minutes = -tm/(60*60) * np.pi * 2
    a_hours = -tm/(60*60*12) * np.pi * 2

    #Define robot geomitry
    #d_seconds = 2.5  
    d_minutes = 2
    d_hours = 1.5 
    arrow_width=0.5
    arrow_length=1

    # Set up figure
    fig = plt.gcf()
    ax = fig.gca();
    ax.set_xlim([-15,15]);
    ax.set_ylim([-10,10]);
    plt.axis('off');
    
    #Define the arrow at the end of the last hand 
    #p = np.matrix([[0,d_minutes,1], [0,0,1]]).T
    p = np.matrix([[0,d_minutes,1], [-arrow_width,d_minutes,1], [0,arrow_length+d_minutes,1], [arrow_width,d_minutes,1 ], [0,d_minutes,1 ], [0,0,1]] ).T;
    
    # Calculation Second hand transformation matrix     
    J2 = np.matrix([[np.cos(a_minutes), -np.sin(a_minutes), 0 ], 
                    [np.sin(a_minutes), np.cos(a_minutes), d_hours ], 
                    [0, 0, 1]])
    p = np.concatenate((J2*p, np.matrix([0,0,1]).T), axis=1 )
    
    J1 = np.matrix([[np.cos(a_hours), -np.sin(a_hours), 0 ], 
                    [np.sin(a_hours), np.cos(a_hours), 0 ], 
                    [0, 0, 1]])
    p = np.concatenate((J1*p, np.matrix([0,0,1]).T), axis=1 )

    ax.scatter(0,0, s=20, facecolors='r', edgecolors='r')
    ax.plot(p[0,:].tolist()[0],(p[1,:]).tolist()[0])

#Run the clock for about 5 hours at 100 times speed so we can see the hands move
for tm in range(0,60*60*5, 100):
    odd_clock(tm);
    show_animation();
# 'Run' this cell to see the animation

Question

Using the given point (\(p\)) written in “minutes” coordinates (on line 26 of the above code) and the above transformation matrices (\(J_1\),\(J_2\)), write down the equation to transform \(p\) into world coordinates \(p_w\).

Question

Notice the above odd_clock function has variables d_seconds and a_seconds commented out. Use these variables and modify the above code to add a “seconds” hand on the tip of the minute hand such that the seconds hand moves around the minute hand just like the minute hand moves around the hour hand. If you have trouble, use the following cell to explain your thought process and where you are getting stuck.